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Modified Schwarz-Christoffel mappings using approximate curve factors
Jönköping University, School of Engineering, JTH, Mathematics.
2009 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 233, no 4, p. 1117-1127Article in journal (Refereed) Published
Abstract [en]

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.

In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C1 regularity, C regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.

Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C) boundary curve can be achieved.

Place, publisher, year, edition, pages
Elsevier , 2009. Vol. 233, no 4, p. 1117-1127
Keywords [en]
Conformal mapping, Schwarz-Christoffel mapping, Approximate curve factor
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-10513DOI: 10.1016/j.cam.2009.09.006OAI: oai:DiVA.org:hj-10513DiVA, id: diva2:241714
Available from: 2009-10-05 Created: 2009-10-05 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Numerical conformal mappings for waveguides
Open this publication in new window or tab >>Numerical conformal mappings for waveguides
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can be re-formulated as a two-dimensional scattering problem, provided that the variations take place in only one dimension at a time. By using the so-called Building Block Method, it is possible to construct the scattering properties of a combination of scatterers when the properties of each scatterer are known. Hence, variations in the waveguide geometry or in the boundary conditions can be treated one at a time.

Using the Building Block Method, the problem takes the form of the Helmholtz equation for stationary waves in a waveguide of infinite length and with smoothly varying geometry and boundary conditions. A conformal mapping is used to transform the problem into a corresponding problem in a straight horizontal waveguide, and by expanding the field in Fourier trigonometric series, the problem can be reformulated as an infinite-dimensional ordinary differential equation. From this, numerically solvable differential equations for the reflection and transmission operators are derived.

To be applicable in the Building Block Method, the numerical conformal mapping must be constructed such that the direction of the boundary curve can be controlled. At the channel ends, it is an indispensable requirement, that the two boundary curves are (at least) asymptotically parallel and straight. Furthermore, to achieve bounded operators in the differential equations, the boundary curves must satisfy different regularity conditions, depending on the boundary conditions.

In this work, several methods to accomplish such conformal mappings are presented. The Schwarz–Christoffel mapping, which is a natural starting point and for which also efficient numerical software exists, can be modified in different ways in order to achieve polygons with rounded corners. We present algorithms by which the parameters in the mappings can be determined after such modifications. We show also how the unmodified Schwarz–Christoffel mapping can be used for regions with a smooth boundary. This is done by constructing an appropriate outer polygon to the considered region.

Finally, we introduce one method that is not Schwarz–Christoffel-related, by showing how one of the so-called zipper algorithms can be used for waveguides.

Place, publisher, year, edition, pages
Växjö: Växjö University Press, 2009. p. 120
Series
Acta Wexionesa, ISSN 1404-4307 ; 177
Keywords
waveguides, building block method, numerical conformal mappings, Schwarz–Christoffel mapping, rounded corners method, approximate curve factors, outer polygon method, boundary curvature, zipper method, geodesic algorithm, acoustic wave scattering, electro-magnetic wave scattering
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-10602 (URN)978-91-7636-661-5 (ISBN)
Public defence
2009-09-25, Weber, Växjö universitet, Växjö, 10:00 (English)
Opponent
Supervisors
Available from: 2009-12-22 Created: 2009-10-10 Last updated: 2009-12-22Bibliographically approved

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